Determination of relative permeability data is required for almost all calculations of fluid flow in petroleum reservoirs. Water-oil relative permeability data play important roles in characterizing the simultaneous two-phase flow in porous rocks and predicting the performance of immiscible displacement processes in oil reservoirs. They are used, among other applications, for determining fluid distortions and residual saturations, predicting future reservoir performance, and estimating ultimate recovery. Undoubtedly, these data are considered among the most valuable information required in reservoir simulation studies.
Estimates of relative permeability are generally obtained from laboratory experiments with reservoir core samples. Because the protocols for laboratory measurement of relative permeability are intricate, expensive and time consuming, empirical correlations are usually used to predict relative permeability data, or to estimate them in the absence of experimental data. However, prior art methodologies for developing empirical correlations for obtaining accurate estimates of relative permeability data have been of limited success and proven difficult, especially for carbonate reservoir rocks. In comparison, clastic reservoir rocks are more homogeneous in terms of pore size, rock fabric and grain size distribution, and therefore have similar pore size distribution and similar flow conduits. This is difficult because carbonate reservoirs are highly heterogeneous due to changes of rock fabric during diagenetic altercation, chemical interaction, the presence of fossil remains and vugs and dolomitization. This complicated rock fabric, different pore size distribution, leads to less predictable different fluid conduits due to the presence of various pore sizes and rock families.
Artificial neural network (ANN) technology has proved successful and useful in solving complex structure and nonlinear problems. ANNs have seen an expansion of interest over the past few years. They are powerful and useful tools for solving practical problems in the petroleum industry, as described by Mohaghegh. S. D. in “Recent Developments in Application of Artificial Intelligence in Petroleum Engineering”, JPT 57 (4): 86-91, SPE-89033-MS, DOI: 10.2118/89033-MS., 2005; and by Al-Fattah, S. M., and Startzman, R. A. in “Neural Network Approach Predicts U.S. Natural Gas Production”, SPEPF 18 (2): 84-91, SPE-82411-PA, DOI: 10.2118/82411-PA, 2003. The disclosures of these articles are incorporated herein by reference in their entirety.
Advantages of neural network techniques over conventional techniques include the ability to address highly nonlinear relationships, independence from assumptions about the distribution of input or output variables, and the ability to address either continuous or categorical data as either inputs or outputs. See, for example, Bishop, C., “Neural Networks for Pattern Recognition”, Oxford: University Press, 1995; Fausett, L., “Fundamentals of Neural Networks”, New York: Prentice-Hall, 1994; Haykin, S., “Neural Networks: A Comprehensive Foundation”, New York: Macmillan Publishing, 1994; and Patterson, D., “Artificial Neural Networks”, Singapore: Prentice Hall, 1996. The disclosures of these articles are incorporated herein by reference in their entirety. In addition, neural networks are intuitively appealing as they are based on crude, low-level models of biological systems. Neural networks, as in biological systems, learn from examples. The neural network user provides representative data and trains the neural networks to learn the structure of the data.
One type of ANN known to the art is the Generalized Regression Neural Network (GRNN) which uses kernel-based approximation to perform regression, and was described in the above articles by Patterson in 1996 and Bishop in 1995. It is one of the so-called Bayesian networks. GRNN have exactly four layers: input layer, radial centers layer, regression nodes layer, and output layer. As shown in FIG. 1, the input layer has an equal number of nodes as input variables. The radial layer nodes represent the centers of clusters of known training data. This layer must be trained by a clustering algorithm such as Sub-sampling, K-means, or Kohonen training. The regression layer, which contains linear nodes, must have exactly one node more than the output layer. There are two types of nodes: the first type of node calculates the conditional regression for each output variable, whereas the second type of node calculates the probability density. The output layer performs a specialized function such that each node simply divides the output of the associated first type node by that of the second type node in the previous layer.
GRNNs can only be used for regression problems. A GRNN trains almost instantly, but tends to be large and slow. Although it is not necessary to have one radial neuron for each training data point, the number still needs to be large. Like the radial basis function (RBF) network, the GRNN does not extrapolate. It is noted that prior applications of the GRNN-type of ANNs have not been used for relative permeability determination.